Modeling of Continuous Time Index Fund and its Statistical Test

本研究关注的是一个连续时间的财务问题，寻找一个类似于基准投资组合的指数基金。将该问题转化为具有均方误差跟踪的最优随机控制问题。本研究的主要结果是对静态情形的一些推广，主要结果如下：(1)如果基准投资组合处于有效前沿，则最优指数基金在任意时间点也处于有效前沿。(2)如果基准投资组合不在有效前沿，则最优指数基金在任何时间点都不在有效前沿。(3)遗传算法是寻找最优指数基金的最有效算法。我们的一个结果发表在讨论论文中，并要求研究人员发表评论和反思。另一篇论文提交给了学术期刊。此外，所有结果都在日本OR学会的研讨会和在韩国举行的国际会议上得到了报告。此外，还在日本出版了两本书，其中包含了我们研究结果的基本概念。

This research is concerned with a continuous time financial problem to find an index fund similar to the benchmark portfolio. The problem is formulated into the optimal stochastic control problem under the mean square error tracking at the final point in time horizon. The main results obtained in this research are some extensions of the static case and described as follows:(1) If the benchmark portfolio is on the efficient frontier, then the optimal index fund is also on the efficient frontier for any point in time.(2) If the benchmark portfolio is not on the efficient frontier, then the optimal index fund is not also on the efficient frontier for any point in time.(3) The genetic algorithm is most efficient to find an optimal index fund.One of our results is published in the discussion paper and the comments and reflects from researchers have been required. The other paper is submitted to the academic journal. Moreover, all the results were reported in the workshop in the OR society of Japan and in the international conference held in Korea. In addition, two books in which include the fundamental concept of our results are published in Japan.